A matter of degree
Statistical evidence for causality combines observed data with a mathematical model of the world
Causal evidence varies in terms of complexity of math/assumptions: a matter of degree
Model-based inferences about causality depend on complex statistical models with many assumptions
Design-based inferences about causality use carefully controlled comparisons with simple, transparent models and assumptions
Whatever our approach…
do the assumptions needed to use this mathematical tool reasonably fit reality?
When we condition, we block specific backdoor paths that generate confounding by comparing cases that are similar on observable trait:
In the absence of an experiment, we might choose more careful research designs, instead of relying on conditioning.
Here, the structure of the comparison motivates an argument for independence of cause and potential outcomes. Rather than block specific confounding variables, we eliminate confounding due to a class of confounding variables.
Social and political theorists have frequently argued that media—by shaping perceptions of events in the world, exposing people to narrative frames—affects beliefs and behaviors.
Foos and Bischoff examine the effect of changing exposure to The Sun on anti-EU attitudes and voting in the UK.
We could ignore this shift, and simply compare attitudes about the EU in areas (or among people) with greater vs less readership of The Sun:
We could instead, compare attitudes toward the EU in Liverpool before and after the boycott:
In interrupted time series, we plug in the observed outcome before treatment for the counterfactual outcome where treatment did not happen: where \(t=1\) indicates post-treatment, \(t=0\) indicates before treatment.
\[\tau_i = [Y_{i,t=1}(1) | D_i = 1] - \color{red}{[Y_{i,t=1}(0)|D_i = 1]}\]
Plugging in:
\[\widehat{\tau_i} = [Y_{i,t=1}(1) | D_i = 1] - [Y_{i,t=0}(0) | D_i = 1]\]
What can go wrong here?
In order for interrupted time series to work, we must assume:
\[[Y_{i,t=0}(0) | D_i = 1] = \color{red}{[Y_{i,t=1}(0)|D_i = 1]}\]
That in the absence of the treatment, outcomes of \(Y\) would not have changed from before to after treatment.
Assumptions mean: none of the following
A good example:
Difference-in-differences compares changes in the treated cases against changes in untreated cases.
Again, we want to know:
\[\begin{split}E[\tau_i | D_i = 1] = {} \frac{1}{n}\sum\limits^{n}_{i=1} & [Y_{i,t=1}(1) | D_i = 1] - \\ & \color{red}{[Y_{i,t=1}(0)|D_i = 1]}\end{split}\]
Interrupted Time Series forced us to assume no other changes in outcomes over time, but it is almost always true that:
\(\color{red}{Y_{i,t=1}(0) | D_i = 1]} - [Y_{i,t=0}(0)|D_i = 1] \neq 0\)
That is to say, counterfactually, in the absence of treatment \(D\), potential outcomes \(Y_i(0)\) are changing over time.Observed pre-treatment outcomes not a good substitute for post-treatment counterfactual outcomes.
We don’t know: \(\color{red}{Y_{i,t=1}(0) | D_i = 1]} - [Y_{i,t=0}(0)|D_i = 1]\)
But we do know: \(Y_{i,t=1}(0) | D_i = 0] - [Y_{i,t=0}(0)|D_i = 0]\)
In our example: we don’t know how EU skepticism might have trended in Liverpool absent the boycott. But we do know how EU skepticism in the rest of the UK trended absent the boycott.
If we assume:
\[\{\overbrace{\color{red}{Y_{i,t=1}(0) | D_i = 1]} - [Y_{i,t=0}(0)|D_i = 1]}^{\text{Treated counterfactual trend}}\} = \\ \{\underbrace{Y_{i,t=1}(0) | D_i = 0] - [Y_{i,t=0}(0)|D_i = 0]}_{\text{Untreated observed trend}}\}\]
Then we can plug-in the \(observed\) untreated group trend for the \(\color{red}{counterfactual}\) treated group trend.
This is the parallel trends assumption. It is equivalent to saying there are no time-varying confounding variables that differ between treated and untreated.
If it is true, we can do some simple algebra and find that
\([\tau_i | D_i = 1] = [Y_{i,t=1}(1) | D_i = 1] - \color{red}{[Y_{i,t=1}(0)|D_i = 1]}\)
\(\begin{equation}\begin{split}[\tau_i | D_i = 1] = {} & \{\overbrace{[Y_{i,t=1}(1) | D_i = 1] - [Y_{i,t=0}(0) | D_i = 1]}^{\text{Treated observed trend}}\} - \\ & \{\underbrace{\color{red}{Y_{i,t=1}(0) | D_i = 1]} - [Y_{i,t=0}(0)|D_i = 1]}_{\text{Treated counterfactual trend}}\}\end{split}\end{equation}\)
\(\begin{equation}\begin{split}[\widehat{\tau_i} | D_i = 1] = {} & \{\overbrace{[Y_{i,t=1}(1) | D_i = 1] - [Y_{i,t=0}(0) | D_i = 1]}^{\text{Treated observed trend}}\} - \\ & \{\underbrace{Y_{i,t=1}(0) | D_i = 0] - [Y_{i,t=0}(0)|D_i = 0]}_{\text{Untreated observed trend}}\}\end{split}\end{equation}\)
And this gives us the name:
This shows that the Boycott of the Sun reduced Euro Skepticism in Liverpool
Under the parallel trends assumption (untreated cases have the same trends as treated cases in the absence of treatment)
This is an unbiased estimate of the Average Treatment on Treated (\(ATT\)):
If parallel trends assumption holds, what kinds of confounding does this design eliminate?
What are examples of confounders held constant in Sun Boycott difference in difference?
Estimation:
How do we validate the parallel trends assumption?
Placebo Tests:
Staggered Treatment: Bacon-Goodman 2021
Multiple Treatments: https://arxiv.org/pdf/1803.08807.pdf
Continuous Treatment: https://psantanna.com/files/Callaway_Goodman-Bacon_SantAnna_2021.pdf
As we get away from simple DiD, assumptions and potential problems multiply, solutions get more complicated…
What was the effect of enlistment in the US Civil War on voting for the Republican party?
\(GOP_{ie} = \alpha_i + \alpha_e + \beta Enlist_i \times PostWar_e + \epsilon_y + \epsilon_i\)
Assumptions:
Caveats:
Address confounding in a different way:
Distinguishing “natural experiment” from experiments:
An observational study where causal inference comes from the design that draws on randomization.
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Two approaches:
Decisions:
Assumptions:
Follows from Wald estimator for non-compliance:
Decisions
Assumptions: